Abstract
We consider locally balanced Gray codes.We say that a Gray code is locally balanced if every “short” subword in its transition sequence contains all letters of the alphabet |1, 2,..., n~. The minimal length of these subwords is the window width of the code. We show that for each n ≥ 3 there exists a Gray code with window width at most n + 3⌊log n⌋.
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References
O. P. Godovykh, “A Study of Uniform Gray Codes,” in Proceedings of the 50th International Student Scientific Conference “Students and Progress in Science and Technology,” Novosibirsk, Russia, April 13–19, 2012 (Novosib. Gos. Univ., Novosibirsk, 2012), p.133.
A. A. Evdokimov, “On Enumeration of Subsets of a Finite Set,” in Methods of Discrete Analysis for Solving Combinatorial Problems, Vol. 34 (Inst. Mat., Novosibirsk, 1980), pp. 8–26.
L. A. Korolenko, “Finding Strongly Uniform Gray Codes,” in Proceedings of the 48th International Student Scientific Conference “Students and Progress in Science and Technology,” Novosibirsk, Russia, April 10–14, 2010 (Novosib. Gos. Univ., Novosibirsk, 2010), p.162.
F. Aguilóand A. Miralles, “On the Frobenius’ Problem of Three Numbers,” in Proceedings of 2005 European Conference on Combinatorics, Graph Theory and Applications, EuroComb 2005, Berlin, Germany, September 5–9, 2005 (DMTCS, Nancy, 2005), p. 317–322.
Yu. Chebiryak and D. Kroening, “Towards a Classification of Hamiltonian Cycles in the 6-Cube,” J. Satisf. BooleanModel. Comput. 4 (1), 57–74 (2008).
I. J. Dejter and A. A. Delgado, “Classes of Hamilton Cycles in the 5-Cube,” J. Combin. Math. Combin. Comput. 61, 81–95 (2007).
T. Feder and C. Subi, “Nearly Tight Bounds on the Number of Hamiltonian Circuits of the Hypercube and Generalizations,” Inform. Process. Lett. 109 (5), 267–272 2009).
L. Goddyn and P. Gvozdjak, “BinaryGray Codes with Long Bit Runs,” Electron. J. Comb. 10 (R27), 1–10 (2003).
H. Haanpää and P. R. J. Ö stergård, “Counting Hamiltonian Cycles in Bipartite Graphs,” Math. Comput. 83 (286), 979–995 (2014).
C. Savage, “A Survey of Combinatorial Gray Codes,” SIAM Rev. 39 (4), 605–629 (1997).
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Original Russian Text © I.S. Bykov, 2016, published in Diskretnyi Analiz i Issledovanie Operatsii, 2016, Vol. 23, No. 1, pp. 51–62.
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Bykov, I.S. On locally balanced gray codes. J. Appl. Ind. Math. 10, 78–85 (2016). https://doi.org/10.1134/S1990478916010099
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DOI: https://doi.org/10.1134/S1990478916010099